written 4.9 years ago by |
G = $80 \ \times \ 10^9 \ N/m^2$
J = $1.8 \ \times \ 10^{-6} \ m^4$
$\theta \ = 2˚ = 2 \times \frac{\pi }{180} \ = \ 0.0349$ rad.
$ - kt \ = \ \frac{GJ}{L} = \frac{80 \ \times \ 10^9 \ \times \ 1.8 \times \ 10^{-6}}{0.6}$
$kt \ = \ 240 \ \times \ 10^3$ Nm/rad
$W_n \ = \ \sqrt{ \frac{Kt}{I}}$
$= \sqrt{ \frac{240 \ \times \ 10^3}{1.65}} = 381.38$ rad/sec
$\frac{\theta}{\theta_{st}} = \frac{1}{\sqrt{(1-r^2)^2 + (2 \xi r)^2}}$ [NO DAMPER]
$\frac{\theta}{\theta_{st}} = \frac{1}{1 – r_1^2}$ and $\frac{\theta}{\theta_{st}} = \frac{1}{r_2^2 – 1} $
Now: $\theta_{st} = \frac{Mo}{Kt}$
$= \frac{4000}{240 \ \times \ 10^3}$
= 0.0166 rad
$\therefore$ $\frac{\theta}{\theta \ st} = \frac{0.0349}{0.0166} = 2.102$
$2.101 = \frac{1}{1-r_1^2}$
$r_1 \ = \ 0.72$
$2.102 \ = \ \frac{1}{r_2^2 -1}$
$r^2 \ = \ 1.2147$
Now, $r_1 \ = \ \frac{w_1}{w_n}$
$0.72 \times 381.38 \ = \ w_1$
$w_1 \ = \ 274.59$ rad/sec
$r_2 \ = \ \frac{w_2}{w_n}$
$w_2 \ = \ 1.2147 \times 381.38$
$w_2 \ = \ 463.26$ rad/sec